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G = C42.299D4order 128 = 27

281st non-split extension by C42 of D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C42.299D4, C4.72- 1+4, C42.433C23, C4.262+ 1+4, C8⋊D420C2, C82D415C2, C4⋊D829C2, D4.Q828C2, C4⋊SD1613C2, Q8⋊Q813C2, D4⋊Q830C2, C4⋊C8.79C22, (C2×C8).75C23, D4.2D429C2, C4⋊C4.190C23, (C2×C4).449C24, (C2×D8).74C22, (C22×C4).526D4, C23.306(C2×D4), C4⋊Q8.327C22, C4.128(C8⋊C22), C4⋊M4(2)⋊10C2, C4.Q8.45C22, (C2×D4).191C23, (C4×D4).129C22, (C2×Q8).179C23, (C4×Q8).126C22, C2.D8.111C22, D4⋊C4.56C22, C4⋊D4.211C22, C41D4.177C22, (C2×C42).906C22, Q8⋊C4.54C22, (C2×SD16).40C22, C22.709(C22×D4), C22⋊Q8.216C22, C2.72(D8⋊C22), (C22×C4).1104C23, C22.26C2424C2, C4.4D4.166C22, (C2×M4(2)).87C22, C42.C2.143C22, C23.36C2315C2, C2.68(C22.31C24), (C2×C4).573(C2×D4), C2.67(C2×C8⋊C22), SmallGroup(128,1983)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.299D4
C1C2C4C2×C4C42C4×D4C23.36C23 — C42.299D4
C1C2C2×C4 — C42.299D4
C1C22C2×C42 — C42.299D4
C1C2C2C2×C4 — C42.299D4

Generators and relations for C42.299D4
 G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=a-1, dad=ab2, cbc-1=a2b-1, dbd=b-1, dcd=a2c3 >

Subgroups: 412 in 196 conjugacy classes, 86 normal (44 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×9], C22, C22 [×12], C8 [×4], C2×C4 [×6], C2×C4 [×16], D4 [×16], Q8 [×4], C23, C23 [×3], C42 [×4], C42, C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×4], M4(2) [×2], D8 [×2], SD16 [×2], C22×C4 [×3], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×D4 [×5], C2×Q8, C2×Q8, C4○D4 [×4], D4⋊C4 [×6], Q8⋊C4 [×2], C4⋊C8 [×2], C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C42, C42⋊C2, C4×D4, C4×D4 [×2], C4×D4 [×2], C4×Q8, C4⋊D4, C4⋊D4 [×2], C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, C422C2, C41D4, C4⋊Q8, C2×M4(2) [×2], C2×D8 [×2], C2×SD16 [×2], C2×C4○D4, C4⋊M4(2), C4⋊D8, C4⋊SD16, D4.2D4 [×2], C8⋊D4 [×2], C82D4 [×2], D4⋊Q8, Q8⋊Q8, D4.Q8 [×2], C23.36C23, C22.26C24, C42.299D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C8⋊C22 [×2], C22×D4, 2+ 1+4, 2- 1+4, C22.31C24, C2×C8⋊C22, D8⋊C22, C42.299D4

Character table of C42.299D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A8B8C8D
 size 11114888222222444888888888
ρ111111111111111111111111111    trivial
ρ211111-1-1-1111111111-1-1-1-1-11111    linear of order 2
ρ3111111-1-1-1-11-1-11-11-111-1-11-11-11    linear of order 2
ρ411111-111-1-11-1-11-11-1-1-111-1-11-11    linear of order 2
ρ511111-1-111111111111-1-111-1-1-1-1    linear of order 2
ρ61111111-1111111111-111-1-1-1-1-1-1    linear of order 2
ρ711111-11-1-1-11-1-11-11-11-11-111-11-1    linear of order 2
ρ8111111-11-1-11-1-11-11-1-11-11-11-11-1    linear of order 2
ρ91111-1-1-11-111-1111-1-1111-1-111-1-1    linear of order 2
ρ101111-111-1-111-1111-1-1-1-1-11111-1-1    linear of order 2
ρ111111-1-11-11-111-11-1-1111-11-1-111-1    linear of order 2
ρ121111-11-111-111-11-1-11-1-11-11-111-1    linear of order 2
ρ131111-1111-111-1111-1-11-1-1-1-1-1-111    linear of order 2
ρ141111-1-1-1-1-111-1111-1-1-11111-1-111    linear of order 2
ρ151111-11-1-11-111-11-1-111-111-11-1-11    linear of order 2
ρ161111-1-1111-111-11-1-11-11-1-111-1-11    linear of order 2
ρ1722222000-2-2-2-2-2-22-22000000000    orthogonal lifted from D4
ρ182222-20002-2-22-2-222-2000000000    orthogonal lifted from D4
ρ192222-2000-22-2-22-2-222000000000    orthogonal lifted from D4
ρ202222200022-222-2-2-2-2000000000    orthogonal lifted from D4
ρ214-4-4400000-40040000000000000    orthogonal lifted from C8⋊C22
ρ224-4-4400000400-40000000000000    orthogonal lifted from C8⋊C22
ρ234-44-4000000-4004000000000000    orthogonal lifted from 2+ 1+4
ρ244-44-4000000400-4000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ2544-4-400004i00-4i00000000000000    complex lifted from D8⋊C22
ρ2644-4-40000-4i004i00000000000000    complex lifted from D8⋊C22

Smallest permutation representation of C42.299D4
On 64 points
Generators in S64
(1 57 5 61)(2 62 6 58)(3 59 7 63)(4 64 8 60)(9 40 13 36)(10 37 14 33)(11 34 15 38)(12 39 16 35)(17 32 21 28)(18 29 22 25)(19 26 23 30)(20 31 24 27)(41 56 45 52)(42 53 46 49)(43 50 47 54)(44 55 48 51)
(1 25 51 12)(2 9 52 30)(3 27 53 14)(4 11 54 32)(5 29 55 16)(6 13 56 26)(7 31 49 10)(8 15 50 28)(17 60 38 47)(18 44 39 57)(19 62 40 41)(20 46 33 59)(21 64 34 43)(22 48 35 61)(23 58 36 45)(24 42 37 63)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)
(1 8)(2 7)(3 6)(4 5)(9 10)(11 16)(12 15)(13 14)(17 39)(18 38)(19 37)(20 36)(21 35)(22 34)(23 33)(24 40)(25 28)(26 27)(29 32)(30 31)(41 63)(42 62)(43 61)(44 60)(45 59)(46 58)(47 57)(48 64)(49 52)(50 51)(53 56)(54 55)

G:=sub<Sym(64)| (1,57,5,61)(2,62,6,58)(3,59,7,63)(4,64,8,60)(9,40,13,36)(10,37,14,33)(11,34,15,38)(12,39,16,35)(17,32,21,28)(18,29,22,25)(19,26,23,30)(20,31,24,27)(41,56,45,52)(42,53,46,49)(43,50,47,54)(44,55,48,51), (1,25,51,12)(2,9,52,30)(3,27,53,14)(4,11,54,32)(5,29,55,16)(6,13,56,26)(7,31,49,10)(8,15,50,28)(17,60,38,47)(18,44,39,57)(19,62,40,41)(20,46,33,59)(21,64,34,43)(22,48,35,61)(23,58,36,45)(24,42,37,63), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,40)(25,28)(26,27)(29,32)(30,31)(41,63)(42,62)(43,61)(44,60)(45,59)(46,58)(47,57)(48,64)(49,52)(50,51)(53,56)(54,55)>;

G:=Group( (1,57,5,61)(2,62,6,58)(3,59,7,63)(4,64,8,60)(9,40,13,36)(10,37,14,33)(11,34,15,38)(12,39,16,35)(17,32,21,28)(18,29,22,25)(19,26,23,30)(20,31,24,27)(41,56,45,52)(42,53,46,49)(43,50,47,54)(44,55,48,51), (1,25,51,12)(2,9,52,30)(3,27,53,14)(4,11,54,32)(5,29,55,16)(6,13,56,26)(7,31,49,10)(8,15,50,28)(17,60,38,47)(18,44,39,57)(19,62,40,41)(20,46,33,59)(21,64,34,43)(22,48,35,61)(23,58,36,45)(24,42,37,63), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,8)(2,7)(3,6)(4,5)(9,10)(11,16)(12,15)(13,14)(17,39)(18,38)(19,37)(20,36)(21,35)(22,34)(23,33)(24,40)(25,28)(26,27)(29,32)(30,31)(41,63)(42,62)(43,61)(44,60)(45,59)(46,58)(47,57)(48,64)(49,52)(50,51)(53,56)(54,55) );

G=PermutationGroup([(1,57,5,61),(2,62,6,58),(3,59,7,63),(4,64,8,60),(9,40,13,36),(10,37,14,33),(11,34,15,38),(12,39,16,35),(17,32,21,28),(18,29,22,25),(19,26,23,30),(20,31,24,27),(41,56,45,52),(42,53,46,49),(43,50,47,54),(44,55,48,51)], [(1,25,51,12),(2,9,52,30),(3,27,53,14),(4,11,54,32),(5,29,55,16),(6,13,56,26),(7,31,49,10),(8,15,50,28),(17,60,38,47),(18,44,39,57),(19,62,40,41),(20,46,33,59),(21,64,34,43),(22,48,35,61),(23,58,36,45),(24,42,37,63)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,8),(2,7),(3,6),(4,5),(9,10),(11,16),(12,15),(13,14),(17,39),(18,38),(19,37),(20,36),(21,35),(22,34),(23,33),(24,40),(25,28),(26,27),(29,32),(30,31),(41,63),(42,62),(43,61),(44,60),(45,59),(46,58),(47,57),(48,64),(49,52),(50,51),(53,56),(54,55)])

Matrix representation of C42.299D4 in GL8(𝔽17)

001600000
16161160000
160000000
00010000
000001600
00001000
000000016
00000010
,
016000000
10000000
16161160000
1502160000
00000400
000013000
00000004
000000130
,
10106100000
11910000
1610000000
150260000
00000010
000000016
000001600
000016000
,
10106100000
11910000
17000000
1412260000
00000010
00000001
00001000
00000100

G:=sub<GL(8,GF(17))| [0,16,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,1,0,0,0,0,0,0,0,16,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0],[0,1,16,15,0,0,0,0,16,0,16,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,16,16,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0],[10,1,16,15,0,0,0,0,10,1,10,0,0,0,0,0,6,9,0,2,0,0,0,0,10,1,0,6,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0],[10,1,1,14,0,0,0,0,10,1,7,12,0,0,0,0,6,9,0,2,0,0,0,0,10,1,0,6,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0] >;

C42.299D4 in GAP, Magma, Sage, TeX

C_4^2._{299}D_4
% in TeX

G:=Group("C4^2.299D4");
// GroupNames label

G:=SmallGroup(128,1983);
// by ID

G=gap.SmallGroup(128,1983);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,891,675,80,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a*b^2,c*b*c^-1=a^2*b^-1,d*b*d=b^-1,d*c*d=a^2*c^3>;
// generators/relations

Export

Character table of C42.299D4 in TeX

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